On the norm groups of Galois -extensions of algebraic number fields

Abstract: Let K be a finite extension of a -adic number field k. By local class field theory there is only a finite number of norm subgroups of the multiplicative group of k that contain the norm group . If X is a subgroup of a group Y, then the interval is the set of subgroups of Y that contain X including X and Y. In the present work we investigate the number of norm groups in the interval for a given finite Galois extension of algebraic number fields. There are finite Galois 2-extensions and Galois extensions of odd degrees such that the corresponding intervals contain only a finite number of norm groups. The main theorem, however, states that for any finite Galois extension of even degree that is not a 2-extension, called -extension, the interval contains infinitely many norm groups. [Copyright &y& Elsevier]